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Journal of Physics A Mathematical and General

Study on dynamical critical exponents of the Ising model using the damage spreading method

F Wang, Naomichi Hatano, Masuo Suzuki

Published Aug 21, 1995• 46 citations
Publisher

Abstract

The dynamical behaviour of the Ising model with heat-bath dynamics is studied. According to our investigation of the model using the damage spreading method, the dynamical phase transition temperature is estimated as 4.52+or-0.03 for the 3D Ising model. The dynamical critical exponent z has been obtained as 2.16+or-0.04 for the 2D ferromagnet and 2.09+or-0.04 for the 3D Ising model. The scaling property of magnetization is obtained.

DOI
10.1088/0305-4470/28/16/012

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Journal of Physics A Mathematical and General•Dec 7, 1993•6 citations
The dynamical behaviour of the Ising model with heat-bath dynamics is studied. According to our investigation of the model using the damage spreading method, the dynamical phase transition temperature is estimated as 4.52+or-0.03 for the 3D Ising model. The dynamical critical exponent z has been obtained as 2.16+or-0.04 for the 2D ferromagnet and 2.09+or-0.04 for the 3D Ising model. The scaling property of magnetization is obtained.
By studying damage spreading in spin models in a temperature gradient, it is possible to determine not only the critical temperature, and the correlation length exponent v, but also a second critical exponent beta with high precision. This is done by studying the dynamics of damage spreading, in particular the overhangs which develop in the damage front. The authors demonstrate the method for the two-dimensional Ising model using heat bath dynamics, and discuss differences with Metropolis dynamics.
The authors use the method of 'damage spreading' to measure the time taken for the equilibrium damage (the magnetization) to disappear in finite systems (10<or=L<or=103) at the critical temperature of the infinite system. This time is found to scale as L z with an exponent z of 2.16+or-0.02. This value is compatible with the expected value of the dynamic exponent for the 2D Ising model, but they argue that this method is unable to measure fluctuations properly which they show are responsible for the higher values recently reported in the literature. They suggest that the slowest mode of relaxation has an exponent of z>or=2.27.
We consider power spectra for fluctuating quantities at a critical point, P( omega ) varies as omega - phi , where phi is determined by the dynamic exponent z and static exponents specific to the quantity considered. From the magnetization spectra obtained with simulations, z is found to be 2.13+or-0.03 for the two-dimensional Glauber kinetic Ising model, hence in accord with recent Monte Carlo relaxational studies, but at odds with those obtained by series expansion and damage spreading methods. Excellent dynamic finite-size scaling for system sizes L=10 to 128 supports our estimate.